This is a note to self on computing the lower bound on number of messages each process in a distributed failure detector must send to guarantee adherence to pre-specified values for:

• Time to first detection of a true failure.
• False positive detection rate.
• Message loss rate.

In other words, the failure detector accepts as parameters the above three values, and for those guarantees to be respected, each node in the failure detector cluster must send at least $m$ number of messages, which is what we are after.

The original paper from which I learnt most of this is On Scalable and Efficient Distributed Failure Detectors.

We shall talk about eventually complete (every failure is eventually detected) failure detection, in a crash-stop failure model (actually it isn’t hard to extend the analysis to a crash-recovery model, but we don’t assume Byzantine failures).

## Parameters

### 1. Time to first detection of a true failure

The failure detector in question accepts a parameter $T$, which is the maximum time in seconds between a process failing, and some other process detecting this. Note that this explicitly does not assume any dissemination mechanism, and depending on what it is, first detection may or may not mean detection by all non-faulty processes.

### 2. False positive detection rate

The next parameter the failure detector accepts is $\alpha_{T}$, the acceptable false failure detection rate till the first true failure detection. This is a bit tricky to understand. Consider at time $t_0$, we have a bunch of non-faulty processes $p_1, p_2, p_3.., p_M$ that have not been marked failed yet. We want the probability that any non faulty process marks any of $p_i$ as failed till $t_0 + T$ to be capped at $\alpha_{T}$. To look at this from another perspective, consider 1000 random time periods, each of length $T$. For each time period, we count the number of false failure detections, and sum them up (good nodes marked as faulty by another good node). We want this number to be no more than $1000\alpha_T$.

### 3. Message loss rate

The application also specifies the probability of any given message being lost. This is applied independently to each message, and hence may not be a true picture of what happens in the real world (correlated message losses around congested network zones), but hey at least we are not talking about a spherical cow in vaccum. We call this parameter $p_{ml}$.

## Result

With the parameters specified as above, in an $N$ node distributed failure detector, the minimum number of messages each node has to send per second is given by

This is neat, because it does not depend on the cluster size $N$!

## Proof

Suppose each node sends $m$ messages out (as heartbeats or pings to other nodes) in a time period $T$. The only way a non-faulty process can appear as faulty to all others is if all its outgoing messages are dropped in the time-period of $T$ seconds. This happens with probability $\left(p_{ml}\right)^m$. We want this probability to be capped at $\alpha_T$. So,

Since $m$ is the number of messages sent every T seconds, the number of messages per second $m_r \ge \frac{1}{T} \frac{\log\alpha_T}{\log{p_{ml}}}$, and hence, $m_{min} = \frac{1}{T} \frac{\log\alpha_T}{\log{p_{ml}}}$.